Using bags and blocks for Problem C7, for example, would have required that we draw "negative" blocks  not an easy task!

Another way to solve the equation in Problem C7 is to think about each side of the equation as a function. Think of creating two functions, one to represent each side of the equation. We'll use the variable y to represent the total for each side of the equation.

Using x and y as variables, the left side of the equation in Problem C7 corresponds to the line equation y = 4x - 2. The right side of the equation corresponds to the line equation y = 5x - 3.

To solve the original equation, we want to find common solutions to both of the new equations. A common solution is called a solution to a system of equations. In this section, we will take a closer look at solving systems of linear equations by graphing.

Graphs are useful tools for visualizing how to solve equations. We will create our own coordinate system, using tape to represent the axes and counters to represent the points of each graph.

First, we will need to create the axes. The x-axis is horizontal, the y-axis is vertical, and the two should intersect in the center of the area we're using. The x-axis should be labeled in increments from -3 to +3, and the y-axis should be labeled in increments from -10 to +10. If you prefer, you may print out this picture of the coordinate system:

Seven counters will make the first graph. Follow these rules:

a.

Line up the counters along the x-axis at each integer value between and including -3 and +3.

b.

At each point, multiply the x value by 2, then subtract 3 from the result. Move the counter vertically to the final result of this calculation, making sure not to move horizontally.